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Three-curves theorems

This post deals with a family of theorems that precise and extend the usual maximum modulus principle for analytic functions. Contrary to my previous post on the Phragmen-Lindelöf principle, we are less interested in the fact that the maximal modulus principle is extended to unbounded domains (although this is indeed the case in several of the following examples) than in giving a quantitative form, which will allow us to say that the parts of the boundary that are closer create a stronger constraint on the modulus of the function. In the following, denotes a complex number, with and two real numbers, its real and imaginary part. I use as a reference Real and complex analysis, second edition, by Walter Rudin.

The three-lines theorem

Let be the open vertical strip in the complex plane defined by

Theorem 1

Let be a function that is holomorphic in , continuous on and bounded on For , we set

Then

Let us note that Theorem 1 states that is a convex function on . To prove this, we will use the following Lemma.

Lemma 1

Let be a function that is holomorphic in , continuous on and bounded on If for all , then for all .

Proof. The above Lemma is a direct consequence of a Phragmen-Lindelöf type result on the domain . However, the hypothesis that is bounded allows us to give a simpler proof, taken from Rudin (Theorem 12.8).

Let us consider . For any , we define the function

For all , , therefore and . On the other hand, for all , , and thus.
We chose such that and . Let us consider the open rectangle defined by

The function is holomorphic in , continuous on , and, for all , . Since is in , we have, according to the maximum modulus principle, . The inequality holds for arbitrary , and tends to as tends to . Therefore, we obtain . QED.

Let us now prove the general case.

First, let us treat the case when , that is to say when is identically on the imaginary axis. We will show that in this case, is identically on , and the inequality is then trivial (by the way, this is an answer to Exercice 7, Chapter 12 in Rudin).

Let us consider the function defined by . We denote by the horizontal strip defined by

We have . We now set . The function is continuous on , holomorphic in , and is identically on the real axis. According to the Schwarz reflexion principle (see for instance Rudin, Theorem 11.17), there exists a function , holomorphic on the strip

such that for all . We know (by taking the limit of as tends to ) that is identically on the real axis. By the isolated zeros theorem, is identically on , and in particular is identically on , and thus is identically on . If , we obtain that is identically on by a symmetry argument: considering brings us back to the previous case.

We now assume that and . We set . The function is entire. We have

for all , and in particular . This implies that has no zero and that is bounded. Furthermore, for all , and . This implies that for all . According to Lemma 2, we have

for all . Thus if , we have, for all ,

This gives us the desired result.

From Theorem 1, we obtain a more general three-line theorem simply by using a linear change of variable (actually, we already used one that to apply Schwartz reflexion principle). Let and be real numbers with and let be the open vertical strip defined by

Theorem 2

Let be a function that is holomorphic in , continuous on and bounded on . For , we set

We have

Proof. We define . We have . We set and apply Theorem 1 to . QED.

In the same way, we can formulate and prove a theorem for horizontal strips. Let and be real numbers with and let be the open horizontal strip defined by

Theorem 3

Let be a function that is holomorphic in , continuous on and bounded on . For , we set

We have

Proof. We use the change of variable and apply Theorem 2. QED.

The three-circles theorem

The theorem is due to Jacques Hadamard. There exist several proofs, here we will deduce it from Theorem 2 (see Rudin, Chapter 12, Exercise 8).

We consider and we denote by the open annulus defined by

Theorem 4

Let be a function that is holomorphic in and continuous on . For , we set

Then, we have

Let us note that in this case, is clearly bounded on , since it is continuous and is compact. Furthermore, the maximum modulus principle for an holomorphic function on a bounded domain tells us that the maximum of is reached on one of the boundary circles and . The improvement resides in the fact that Theorem 4 is quantitative: it tells us for instance that if we are closer to the inner circle, the maximum of the modulus of on this circle has more weight in controlling the modulus of .

Let us also note that Theorem 4 can be expressed as the fact that is a convex function of . Indeed, it was first stated in this form by Hadamard.

Proof. The exponential function maps the open vertical strip to and its closure to (the mapping is not one-to-one, but it doesn’t matter here). We set and apply Theorem 2 to . QED.

The three-rays theorem

I haven’t found this last result stated anywhere in quite this way, but it a straightforward transposition of Theorem 3. Let be a number in and another number such that .
We define the open sector by

Theorem 5

Let be a function that is holomorphic in , continuous and bounded on . For , we set

Then, we have

Proof. The exponential function sends the horizontal strip to the open sector . The closure of this strip is sent to the closure of the sector minus the origin. We set and apply Theorem 3 to . QED.